Antonia Ellerbrock

Daniel Blankenburg, Antonia Ellerbrock, Thomas Kesselheim et al.

We study the problem of minimizing an ordered norm of a load vector (indexed by a set of $d$ resources), where a finite number $n$ of customers $c$ contribute to the load of each resource by choosing a solution $x_c$ in a convex set $X_c \subseteq \mathbb{R}^d_{\geq 0}$; so we minimize $||\sum_{c}x_c||$ for some fixed ordered norm $||\cdot||$. We devise a randomized algorithm that computes a $(1+\varepsilon)$-approximate solution to this problem and makes, with high probability, $\mathcal{O}\left((n+d) (\varepsilon^{-2}+\log\log d)\log (n+d)\right)$ calls to oracles that minimize linear functions (with non-negative coefficients) over $X_c$. While this has been known for the $\ell_{\infty}$ norm via the multiplicative weights update method, existing proof techniques do not extend to arbitrary ordered norms. Our algorithm uses a resource price mechanism that is motivated by the follow-the-regularized-leader paradigm, and is expressed by smooth approximations of ordered norms. We need and show that these have non-trivial stability properties, which may be of independent interest. For each customer, we define dynamic cost budgets, which evolve throughout the algorithm, to determine the allowed step sizes. This leads to non-uniform updates and may even reject certain oracle solutions. Using non-uniform sampling together with a martingale argument, we can guarantee sufficient expected progress in each iteration, and thus bound the total number of oracle calls with high probability.

Daniel Ebert, Antonia Ellerbrock

We extend the list of games where the nucleolus is computable in polynomial time. Based on the classical MPS scheme, nucleolus computation can be reduced to the problem of finding a coalition with minimum excess that does not belong to a given linear subspace. We call this problem LSA-MinExcess, and show that it is equivalent to NZ-MinExcess: Given integral values per player, find a coalition with minimum excess whose player values do not sum up to $0$. Exploiting this representation, we prove that the nucleolus is computable in polynomial time for arboricity games, network strength games, and certain $b$-matching games. Along these lines, we show that for $b$-matching games with $b \leq 2$, LSA-MinExcess is polynomially equivalent to Shortest Non-Zero Cycle. Further, we prove that in general, linear subspace avoidance strictly increases the complexity of the minimum excess problem, even for monotone games. We still provide a reduction that trades linear subspace avoidance against an arbitrarily small approximation error. Finally, we show that the nucleolus is unstable in the following sense: A small change in the value function of the game can lead to a change in the nucleolus that is exponential in the number of players.